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  "body": "\n## Statement\n\nLet $X$ be a set with two [[binary operations|0M418FE7BB]] $\\circ$ and $\\otimes$.\nAssume that $\\circ$ has an [[identity element|07WYQPB8E3]] $1_\\circ$, that $\\otimes$ has an identity element $1_\\otimes$, and that the interchange law\n$$(a \\otimes b) \\circ (c \\otimes d) = (a \\circ c) \\otimes (b \\circ d)$$\nholds for all $a, b, c, d \\in X$.\nThen $1_\\circ = 1_\\otimes$, the two operations are equal, and this common operation is commutative and associative.\n\n## Remarks\n\nPut $a = d = 1_\\circ$ and $b = c = 1_\\otimes$ in the interchange law: the left side is $1_\\circ \\circ 1_\\circ = 1_\\circ$ and the right side is $1_\\otimes \\otimes 1_\\otimes = 1_\\otimes$, so the units agree; write $1$ for both.\nThen $a \\circ d = (a \\otimes 1) \\circ (1 \\otimes d) = (a \\circ 1) \\otimes (1 \\circ d) = a \\otimes d$, so the operations agree.\nAlso $b \\circ c = (1 \\otimes b) \\circ (c \\otimes 1) = (1 \\circ c) \\otimes (b \\circ 1) = c \\otimes b = c \\circ b$, which is commutativity.\nAssociativity follows in the same way from $(a \\otimes b) \\circ (1 \\otimes c) = (a \\circ 1) \\otimes (b \\circ c)$.\nThe units are needed: on a set with at least two elements, the operations $a \\circ b = a$ and $a \\otimes b = b$ satisfy the interchange law, but they differ.\nThe main application: for $n \\ge 2$, the homotopy group $\\pi_n(X, x_0)$ carries two compatible products, concatenation in two different coordinates, so it is abelian.\n\n## Explanations\n\n### L1 human\n\nSuppose a set has two ways of combining elements, each with a neutral element, and the two ways commute with each other in the sense of the interchange law.\nThen they are secretly one and the same operation, and this operation is commutative.\nA picture: arrange four elements in a square grid.\nCombine them first by rows and then by columns, or first by columns and then by rows; the interchange law says the results agree.\n\n### L2 human\n\nThe argument explains why the higher homotopy groups $\\pi_n$ for $n \\ge 2$ are abelian, while $\\pi_1$ need not be.\nIn dimension $n \\ge 2$ there are two independent directions in which to concatenate maps of cubes.\nFor the same reason, the fundamental group of an H-space, such as a topological group, is abelian.\nThere, pointwise multiplication of loops satisfies the interchange law with concatenation.\nIn category theory it shows that a monoid object in the category of monoids is a commutative monoid.\nIt also shows that the endomorphisms of the unit object in a monoidal category form a commutative monoid.\n",
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    "name": "Eckmann–Hilton argument",
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  "name": "Eckmann–Hilton argument",
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